Intro: Ichiro Suzuki, the Mariners' Right Fielder has played 8 seasons in the MLB (2001-2008), and in all of those seasons he has hit more than 200 hits, a feat no other player has done before. But this year (2009) he missed 8 games early in the season, which puts in jeopardy him acheiving the goal again. That's why I built a probabilistic model of the phenomenon that Ichiro accumulates 200 hits or more during 2009.
The model: For each of the remaining games (Remaining Games = 162 - IchiroGames(8) - 8), which right now (April 24) stands at 146, I simulate the number of possible at bats Ichiro might have. For this I'm using a discrete distribution based upon last year's distribution of at bats per game, which is the following:
Where 4 AB per game is the most frequent, but it allows for different number of AB per game (from 2 to 6). As a second step, I simulate the number of hits he might have on each game. For this I use a binomial distribution. The binomial distribution counts the number of "success events" out of n trials. In this case, a hit is a successful event while an at-bat is a trial. The binomial distribution has 2 parameters which are n=number of trials, and p=probability of a successful event. n is random and comes from the step before (the discrete distribution) but p is fixed. I will update this model frequently with 2 options for p. The first scenario for p will be to use his actual 2009 batting average as an estimate of p. The second approach will be to use his career batting average.
Then I will run 5,000 iterations (like if I had 5,000 different "whole seasons") in which for each one of those I will have a different and random amount of total hits. We can estimate the probability of him reaching 200 hits, simply by counting on how many of the iterations he actually exceeded 200 hits. Simple, right?
Results: This is a histogram of the iterations, on red you'll see the "career" model, which uses a parameter for p=.330, i.e. his career batting averge. The purple histogram is the "actual" model, which uses a parameter for p=.303, his actual batting average (2009).
You'll notice that under the "current" model, his probability of reaching at least 200 hits = 43% (the right hand tail), you can't see the right tail for the red distribution, but the probability for the "career" model = 89%. The histogram also shows the high variability in the total hits, under both models, due to the fact that the season is very young (from 185 to 260 in the RED model). But the expected number of hits for the purple model = 198 hits (just below 200), and for the red model = 215.
So now you know your odds if you want to bet for it. Will you bet on it?
Update: The following chart shows how the probability has changed in time for both models. As of June 10, it's practically a certainty that Ichiro will reach at least 200 hits. The expected total hits for the current model is 238.
